OK, I found the problem. Thanks so much Dave and Brian for your comments, it was a big help.
Last night I removed the side engine panels and opened the hood. Then I removed the instrument panel, and unscrewed the tach cable at the top. Then I started the engine, and the cable did NOT turn. So I disconnected the cable at the bottom (side of engine block right under the exhaust manifold) and sure enough, the cable was broken. It had broke about 4 inches from the bottom.
I phoned my dealer today, and he is ordering the part. Hopefully, the new one will not break....